# Axiom:Axiom of Foundation

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## Axiom

For all non-empty sets, there is an element of the set that shares no element with the set.

That is:

- $\forall S: \paren {\paren {\exists x: x \in S} \implies \exists y \in S: \forall z \in S: \neg \paren {z \in y} }$

The antecedent states that $S$ is not empty.

## Also defined as

It can also be stated as:

- For every non-empty set $S$, there exists an element $x \in S$ such that $x$ and $S$ are disjoint.

- A set contains no infinitely descending (membership) sequence.

- A set contains a (membership) minimal element.

- The membership relation is a strictly well-founded relation on any non-empty set.

## Also known as

The **axiom of foundation** is also known as the **axiom of regularity**.

## Sources

- 1982: Alan G. Hamilton:
*Numbers, Sets and Axioms*... (previous): $\S 4$: Set Theory: $4.2$ The Zermelo-Fraenkel axioms: $\text {ZF9}$ - Weisstein, Eric W. "Zermelo-Fraenkel Axioms." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html